The combination of higher-order modulation formats and coherent detection with digital signal processing enables high spectral efficiency transmission over long-haul optical links. The digital signal processing in a coherent receiver typically comprises a cascade of several blocks, including a chromatic dispersion (CD) compensation block followed by an adaptive butterfly equalizer. The adaptive butterfly equalizer performs polarization de-multiplexing and dynamic compensation of polarization effects (e.g., polarization mode dispersion (PMD) and polarization dependent loss (PDL)).